I want to know if it is enough for me to give dawah (proselytizing or preaching) to those Muslims who are not practicing Islam fully?
Is it wrong if I don't give dawah to a single non Muslim?
How important is giving dawah to non Muslims?
From Nahj al-Balaqa Vol. 2 Pg. 78
Imam Ali: کونو دعاه الناس بغیر السنتکم
Invite people without your tongues. ( In similar narrations it has been said with your deeds or silently)
So if you do that it that way, you can invite everyone to Islam. If you're nice to your neighbor they'll realize that by your looks and ladies observing hijab or your name, you don't need to tell them hey I'm a Muslim...
Having that said if you had a chance to choose between who to teach and how not to, there's no straightforward answer, but rather some suggestions:
Likely your brother in faith is the most important qualification. I honestly don't know which of the other 3 you should choose, I mean at the time of the prophet there were people who converted after a 1 hour discussion...died before doing their first prayer.
I personally think you should go with the person who seeks your guidance, I mean he's asking you questions and enjoys your answers. Some people are surrounded with doubts, but never even care to do a simple google search to find their answers, others bombard their friends, scholars with every new question so they become doubt-free.
Before doing anything seek divine aid from Allah as he's the guider the turner of hearts.
إِنَّكَ لَا تَهْدِي مَنْ أَحْبَبْتَ وَلَٰكِنَّ اللَّهَ يَهْدِي مَن يَشَاءُ ۚ وَهُوَ أَعْلَمُ بِالْمُهْتَدِينَ
Surely you do not guide whomever you love, but Allah guides whomever He decides, and He knows best the ones (who are) rightly-guided.
I am a preacher for all. Whether Muslim, Christian, atheist, agnostic, Sikh, Hindu, etc...
The Quran states:
In the name of Allah, the Beneficent, the Merciful
Praise be to Allah Who hath revealed the Scripture unto His slave, and hath not placed therein any crookedness, (18:1) (He hath made it) Straight (and Clear) in order that He may warn (the godless) of a terrible Punishment from Him, and that He may give Glad Tidings to the Believers who work righteous deeds, that they shall have a goodly Reward,(18:2) Wherein they will abide for ever; (18:3) And to warn those who say: Allah hath chosen a son, (18:4) (A thing) whereof they have no knowledge, nor (had) their fathers, Dreadful is the word that cometh out of their mouths. They speak naught but a lie. (18:5)
From this short passage of surah Kahf you can see 3 groups mentioned. This short passage says that the Quran was revealed to the prophet (pbuh) so Allah could give warning to godless, and at the same time give glad tiding to believers and to warn the Christians.
This is only a short passage of the Quran. The Quran is full of passages and verse to atheist, Christian, Jews, Muslims, idolaters, etc...
Therefore the prophet (pbuh) preached everyone as did the Sahaba this is in the Quran and hadith. Muhammad (pbuh) is told this is one of the main reasons the Quran has been revealed:
And We have revealed the Scripture unto thee only that thou mayst explain unto them that wherein they differ, and (as) a guidance and a mercy for a people who believe. (16:64)
The Quran is for everyone, islam is for everyone. We must not become like the Jews who said "only for us" and become greedy with the scripture. This is mentioned in the Quran:
The similitude of those who were charged with the (obligations of the) Mosaic Law, but who subsequently failed in those (obligations), is that of a donkey which carries huge tomes (but understands them not). Evil is the similitude of people wh o falsify the Signs of God: and God guides not people who do wrong. (62:5)
Many times I have asked Muslims to join me, and go in the town centre to preach to everyone whether they are Christians, atheist, Muslims, etc... The reply of many Muslims is that we must first rectify the Muslim ummah which is at a critical state before preaching the non-Muslims. And they story they usually give is that when they go to non-Muslims these non-Muslims say "we know certain Muslim that is drinking alcohol or gambling or not praying, so why do you not preach your our people first". I have preached for over 4 years to non-Muslims, never have they said anything along these lines rather it is the opposite, it is the Muslims that I preach that usually say why are you preaching us Muslims, we are already Muslims therefore preach the non-Muslims.
The mistake here is not that some Muslims only preach Muslims or non-Muslim, but that Muslims should say that Muslim as a whole should only preach either Muslims or non-Muslims and this is what the title of this question is suggesting "Is it enough to give dawah only to not fully practicing Muslims and not to non Muslims?"
One thing of this question that must be corrected is that da'wah means to invite or call non-Muslims to islam, the better word to use here is islah, which means to reform or improve.
The Quran tells us to invite people to islam:
Invite (all) to the Way of thy Lord with wisdom and beautiful preaching; and argue with them in ways that are best and most gracious: for thy Lord knoweth best, who have strayed from His Path, and who receive guidance. (16:25)